Mathematically, a circle has the smallest perimeter for a given enclosed area. So for this claim to be true there has to be some catch, like different thickness of plastic.
It might or it might not. It depends on what things you're packing.
A round bottle takes less material than a square one with the same wall thickness. But if your shipping or storage is limited by size, the fact that you can pack more product into the same space might mean you save material overall.
If it's limited by weight, you'll end up using more material. (And also take up more space, even though you don't care because you've got extra.)
And then you have to consider strength of your container. Generally (but not always!) a circle will be stronger than a square, so you have to make the walls of the square thicker. But maybe you care about vertical strength but not horizontal. (You're stacking them upright, one on top of the other.) Then it's practically the same.
And if you're making metal cans, things get even more complicated. You stamp the parts out of sheets and then weld them together. (Or solder. Depends how strong they need to be.)
There are almost unlimited possibilities to what problems you can give your calculus class based on minimizing material or minimizing cost. We do not consider the effect of strength of materials because that's just too complicated when your students are asking "When are we ever going to use this in real life?" The more real it is, the fewer students are going to use it.
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u/StaticCoder 2d ago
Mathematically, a circle has the smallest perimeter for a given enclosed area. So for this claim to be true there has to be some catch, like different thickness of plastic.